2024
|
Karimi-Pour, Atiyeh; Trenn, Stephan Funnel control for impulsive switched systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). @inproceedings{KariTren24,
title = {Funnel control for impulsive switched systems},
author = {Atiyeh Karimi-Pour and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/09/Preprint-KT240913.pdf, Preprint},
year = {2024},
date = {2024-12-16},
urldate = {2024-12-16},
booktitle = {Proc. 63rd IEEE Conf. Decision Control (CDC 2024)},
address = {Milan, Italy},
organization = {IEEE},
abstract = {Impulsive switched systems encompass various modes, each exhibiting distinct behaviours. Typically, a switching sequence orchestrates transitions between these modes, where state jumps may occur, potentially undermining output tracking performance or system stability. This work introduces a funnel controller tailored for relative degree one nonlinear impulsive switched systems. Notably, this controller operates solely based on system output without necessitating knowledge of system dynamics. Unlike classical funnel controllers with fixed boundaries, the proposed method dynamically adjusts the funnel boundary for each approaching jump, aiming to preserve adherence to the original boundary. No precise knowledge of jump instances or maps is required; approximate jump intervals and an upper bound for maximum jump height suffice. Theoretical analysis establishes that the error remains within the funnel, facilitating successful reference signal tracking. Performance validation is demonstrated via numerical simulation.},
note = {to appear},
keywords = {funnel-control, nonlinear, relative-degree, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
Impulsive switched systems encompass various modes, each exhibiting distinct behaviours. Typically, a switching sequence orchestrates transitions between these modes, where state jumps may occur, potentially undermining output tracking performance or system stability. This work introduces a funnel controller tailored for relative degree one nonlinear impulsive switched systems. Notably, this controller operates solely based on system output without necessitating knowledge of system dynamics. Unlike classical funnel controllers with fixed boundaries, the proposed method dynamically adjusts the funnel boundary for each approaching jump, aiming to preserve adherence to the original boundary. No precise knowledge of jump instances or maps is required; approximate jump intervals and an upper bound for maximum jump height suffice. Theoretical analysis establishes that the error remains within the funnel, facilitating successful reference signal tracking. Performance validation is demonstrated via numerical simulation. |
Lanza, Lukas; Dennstädt, Dario; Worthmann, Karl; Schmitz, Philipp; Sen, Gökcen Devlet; Trenn, Stephan; Schaller, Manuel Sampled-data funnel control and its use for safe continual learning Journal Article In: Systems & Control Letters, vol. 192, no. 105892, pp. 1-12, 2024, (open access). @article{LanzDann24,
title = {Sampled-data funnel control and its use for safe continual learning},
author = {Lukas Lanza and Dario Dennstädt and Karl Worthmann and Philipp Schmitz and Gökcen Devlet Sen and Stephan Trenn and Manuel Schaller},
url = {https://stephantrenn.net/wp-content/uploads/2024/09/LanzDenn24.pdf, Paper
https://arxiv.org/abs/2303.00523, arXiv},
doi = {10.1016/j.sysconle.2024.105892},
year = {2024},
date = {2024-08-09},
urldate = {2024-08-09},
journal = {Systems & Control Letters},
volume = {192},
number = {105892},
pages = {1-12},
abstract = {We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety- critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.’s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design.},
note = {open access},
keywords = {discrete-time, funnel-control, input-constraints, relative-degree},
pubstate = {published},
tppubtype = {article}
}
We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety- critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.’s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design. |
Berger, Thomas; Hackl, Christoph M.; Trenn, Stephan Asymptotic tracking by funnel control with internal models Proceedings Article In: Proc. European Control Conference (ECC24), pp. 1776-1781, IEEE, Stockholm, Sweden, 2024. @inproceedings{BergHack24,
title = {Asymptotic tracking by funnel control with internal models},
author = {Thomas Berger and Christoph M. Hackl and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/03/Preprint-BHT240326.pdf, Preprint
https://arxiv.org/abs/2310.15544, arXiv},
doi = {10.23919/ECC64448.2024.10590783},
year = {2024},
date = {2024-06-25},
urldate = {2024-06-25},
booktitle = {Proc. European Control Conference (ECC24)},
pages = {1776-1781},
publisher = {IEEE},
address = {Stockholm, Sweden},
abstract = {Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters).},
keywords = {funnel-control, relative-degree, stability},
pubstate = {published},
tppubtype = {inproceedings}
}
Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters). |
2023
|
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Journal Article In: Automatica, vol. 156, no. 111204, pp. 1-10, 2023, (open access). @article{LeeBerg23,
title = {Edge-wise funnel output synchronization of heterogeneous agents with relative degree one},
author = {Jin Gyu Lee and Thomas Berger and Stephan Trenn and Hyungbo Shim},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/LeeBerg23.pdf, Paper
https://arxiv.org/abs/2110.05330, ArXiV},
doi = {10.1016/j.automatica.2023.111204},
year = {2023},
date = {2023-10-01},
urldate = {2023-10-01},
journal = {Automatica},
volume = {156},
number = {111204},
pages = {1-10},
abstract = {When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes.},
note = {open access},
keywords = {funnel-control, networks, nonlinear, relative-degree, synchronization},
pubstate = {published},
tppubtype = {article}
}
When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes. |
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin A novel two stages funnel controller limiting the error derivative Journal Article In: Systems & Control Letters, vol. 179, no. 105601, pp. 1-10, 2023, (open access). @article{HuTren23,
title = {A novel two stages funnel controller limiting the error derivative},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/HuTren23.pdf, Paper},
doi = {10.1016/j.sysconle.2023.105601},
year = {2023},
date = {2023-09-01},
urldate = {2023-09-01},
journal = {Systems & Control Letters},
volume = {179},
number = {105601},
pages = {1-10},
abstract = {As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme to solve the output-tracking control problem for uncertain nonlinear systems with relative degree one and stable internal dynamics. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel.},
note = {open access},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme to solve the output-tracking control problem for uncertain nonlinear systems with relative degree one and stable internal dynamics. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel. |
2022
|
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin Funnel control for relative degree one nonlinear systems with input saturation Proceedings Article In: Proceedings of the 2022 European Control Conference (ECC), pp. 227-232, London, 2022. @inproceedings{HuTren22,
title = {Funnel control for relative degree one nonlinear systems with input saturation},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-HTZ220329.pdf, Preprint},
doi = {10.23919/ECC55457.2022.9837979},
year = {2022},
date = {2022-07-12},
urldate = {2022-07-12},
booktitle = {Proceedings of the 2022 European Control Conference (ECC)},
pages = {227-232},
address = {London},
abstract = {The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly. |
Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Journal Article In: Automatica, vol. 141, no. 110276, pp. 13, 2022, (open access). @article{LeeTren22,
title = {Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling},
author = {Jin Gyu Lee and Stephan Trenn and Hyungbo Shim},
url = {https://stephantrenn.net/wp-content/uploads/2022/08/LeeTren22.pdf, Paper
https://arxiv.org/abs/2012.14580, Extended ArXiv-version},
doi = {10.1016/j.automatica.2022.110276},
year = {2022},
date = {2022-07-01},
urldate = {2022-07-01},
journal = {Automatica},
volume = {141},
number = {110276},
pages = {13},
abstract = {In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying (undirected) graph structure. The proposed coupling law is motivated by the so-called funnel control method studied in adaptive control under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by a high-gain coupling; consequently we call our novel synchronization method ‘(node-wise) funnel coupling.’ By adjusting the conventional proof technique in the funnel control study, we are even able to obtain asymptotic synchronization with the same funnel coupling law. Moreover, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, is analyzed in this paper. In particular, we introduce a single scalar dynamics called ‘emergent dynamics’ which describes the emergent synchronized behavior of the multi-agent system under funnel coupling. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. We illustrate this idea via the example of a distributed median solver based on funnel coupling.},
note = {open access},
keywords = {funnel-control, nonlinear, synchronization},
pubstate = {published},
tppubtype = {article}
}
In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying (undirected) graph structure. The proposed coupling law is motivated by the so-called funnel control method studied in adaptive control under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by a high-gain coupling; consequently we call our novel synchronization method ‘(node-wise) funnel coupling.’ By adjusting the conventional proof technique in the funnel control study, we are even able to obtain asymptotic synchronization with the same funnel coupling law. Moreover, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, is analyzed in this paper. In particular, we introduce a single scalar dynamics called ‘emergent dynamics’ which describes the emergent synchronized behavior of the multi-agent system under funnel coupling. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. We illustrate this idea via the example of a distributed median solver based on funnel coupling. |
2020
|
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Proceedings Article In: Proc. European Control Conference (ECC 2020), pp. 911-916, Saint Petersburg, Russia, 2020. @inproceedings{LeeBerg20,
title = {Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization},
author = {Jin Gyu Lee and Thomas Berger and Stephan Trenn and Hyungbo Shim},
url = {https://stephantrenn.net/wp-content/uploads/2020/02/Preprint-LBTS200204.pdf, Preprint},
doi = {10.23919/ECC51009.2020.9143983},
year = {2020},
date = {2020-05-14},
booktitle = {Proc. European Control Conference (ECC 2020)},
pages = {911-916},
address = {Saint Petersburg, Russia},
abstract = {A new approach to distributed consensus optimization is studied in this paper. The cost function to be minimized is a sum of local cost functions which are not necessarily convex as long as their sum is convex. This benefit is obtained from a recent observation that, with a large gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields. However, design of the large coupling gain requires global information such as network structure and individual agent dynamics. In this paper, we employ a nonlinear time-varying coupling of diffusive type, which we call `edge-wise funnel coupling.' This idea is borrowed from adaptive control, which enables decentralized design of distributed optimizers without knowledge of global information. Remarkably, without a common internal model, each agent achieves asymptotic consensus to the optimal solution of the global cost. We illustrate this result by a network that asymptotically finds the least-squares solution of a linear equation in a distributed manner.},
keywords = {funnel-control, networks, nonlinear, synchronization},
pubstate = {published},
tppubtype = {inproceedings}
}
A new approach to distributed consensus optimization is studied in this paper. The cost function to be minimized is a sum of local cost functions which are not necessarily convex as long as their sum is convex. This benefit is obtained from a recent observation that, with a large gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields. However, design of the large coupling gain requires global information such as network structure and individual agent dynamics. In this paper, we employ a nonlinear time-varying coupling of diffusive type, which we call `edge-wise funnel coupling.' This idea is borrowed from adaptive control, which enables decentralized design of distributed optimizers without knowledge of global information. Remarkably, without a common internal model, each agent achieves asymptotic consensus to the optimal solution of the global cost. We illustrate this result by a network that asymptotically finds the least-squares solution of a linear equation in a distributed manner. |
2019
|
Lee, Jin Gyu; Trenn, Stephan Asymptotic tracking via funnel control Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 4228-4233, Nice, France, 2019. @inproceedings{LeeTren19,
title = {Asymptotic tracking via funnel control},
author = {Jin Gyu Lee and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-LT190910.pdf, Preprint},
doi = {10.1109/CDC40024.2019.9030274},
year = {2019},
date = {2019-12-13},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
pages = {4228-4233},
address = {Nice, France},
abstract = {Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowledge how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control – without any further assumptions – can achieve asymptotic tracking.},
keywords = {funnel-control},
pubstate = {published},
tppubtype = {inproceedings}
}
Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowledge how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control – without any further assumptions – can achieve asymptotic tracking. |
Trenn, Stephan Asymptotic tracking with funnel control Proceedings Article In: PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2019, (online). @inproceedings{Tren19,
title = {Asymptotic tracking with funnel control},
author = {Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/11/45-PAMM19-201900071.pdf, Paper},
doi = {10.1002/pamm.201900071},
year = {2019},
date = {2019-09-09},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
journal = {PAMM - Proc. Appl. Math. Mech.},
publisher = {WILEY-VCH Verlag},
abstract = {Funnel control is a strikingly simple control technique to ensure model free practical tracking for quite general nonlinear systems. It has its origin in the adaptive control theory, in particular, it is based on the principle of high gain feedback control. The key idea of funnel control is to chose the feedback gain large when the tracking error approaches the prespecified error tolerance (the funnel boundary). It was long believed that it is a theoretical limitation of funnel control not being able to achieve asymptotic tracking, however, in this contribution it will be shown that this is not the case.},
note = {online},
keywords = {funnel-control, stability},
pubstate = {published},
tppubtype = {inproceedings}
}
Funnel control is a strikingly simple control technique to ensure model free practical tracking for quite general nonlinear systems. It has its origin in the adaptive control theory, in particular, it is based on the principle of high gain feedback control. The key idea of funnel control is to chose the feedback gain large when the tracking error approaches the prespecified error tolerance (the funnel boundary). It was long believed that it is a theoretical limitation of funnel control not being able to achieve asymptotic tracking, however, in this contribution it will be shown that this is not the case. |
2017
|
Trenn, Stephan Edge-wise funnel synchronization Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 821 - 822, WILEY-VCH Verlag, 2017, ISSN: 1617-7061. @inproceedings{Tren17,
title = {Edge-wise funnel synchronization},
author = {Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre170523.pdf, Preprint},
doi = {10.1002/pamm.201710378},
issn = {1617-7061},
year = {2017},
date = {2017-06-01},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
volume = {17},
number = {1},
pages = {821 - 822},
publisher = {WILEY-VCH Verlag},
abstract = {Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory.},
keywords = {funnel-control, networks, nonlinear, synchronization},
pubstate = {published},
tppubtype = {inproceedings}
}
Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory. |
2015
|
Shim, Hyungbo; Trenn, Stephan A preliminary result on synchronization of heterogeneous agents via funnel control Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2229–2234, 2015. @inproceedings{ShimTren15,
title = {A preliminary result on synchronization of heterogeneous agents via funnel control},
author = {Hyungbo Shim and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-ST150902.pdf, Preprint},
doi = {10.1109/CDC.2015.7402538},
year = {2015},
date = {2015-12-01},
booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan},
pages = {2229--2234},
abstract = {We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research.},
keywords = {funnel-control, networks, nonlinear, stability, synchronization},
pubstate = {published},
tppubtype = {inproceedings}
}
We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research. |
2013
|
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article In: IEEE Trans. Autom. Control, vol. 58, no. 12, pp. 3126–3141, 2013. @article{LibeTren13b,
title = {The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130702.pdf, Preprint},
doi = {10.1109/TAC.2013.2277631},
year = {2013},
date = {2013-08-16},
journal = {IEEE Trans. Autom. Control},
volume = {58},
number = {12},
pages = {3126--3141},
abstract = {The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions. |
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller: time delays and case study Proceedings Article In: Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 1669–1674, 2013. @inproceedings{LibeTren13a,
title = {The bang-bang funnel controller: time delays and case study},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130320.pdf, Preprint
http://ieeexplore.ieee.org/document/6669120, IEEE Xplore Article Number 6669120},
year = {2013},
date = {2013-07-01},
booktitle = {Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland},
pages = {1669--1674},
abstract = {We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered. |
Hackl, Christoph M.; Hopfe, Norman; Ilchmann, Achim; Mueller, Markus; Trenn, Stephan Funnel control for systems with relative degree two Journal Article In: SIAM J. Control Optim., vol. 51, no. 2, pp. 965–995, 2013. @article{HackHopf13,
title = {Funnel control for systems with relative degree two},
author = {Christoph M. Hackl and Norman Hopfe and Achim Ilchmann and Markus Mueller and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/HackHopf13.pdf, Paper},
doi = {10.1137/100799903 },
year = {2013},
date = {2013-03-19},
journal = {SIAM J. Control Optim.},
volume = {51},
number = {2},
pages = {965--995},
abstract = {Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. |
2012
|
Hackl, Christoph M.; Trenn, Stephan The bang-bang funnel controller: An experimental verification Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 735–736, GAMM Annual Meeting 2012, Darmstadt Wiley-VCH Verlag GmbH, Weinheim, 2012. @inproceedings{HackTren12,
title = {The bang-bang funnel controller: An experimental verification},
author = {Christoph M. Hackl and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-HT120427.pdf, Preprint},
doi = {10.1002/pamm.201210356},
year = {2012},
date = {2012-03-01},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
volume = {12},
number = {1},
pages = {735--736},
publisher = {Wiley-VCH Verlag GmbH},
address = {Weinheim},
organization = {GAMM Annual Meeting 2012, Darmstadt},
abstract = {We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive. |
2010
|
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller Proceedings Article In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. @inproceedings{LibeTren10,
title = {The bang-bang funnel controller},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806.pdf, Preprint
http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806longVersion.pdf, Preprint (long version)},
doi = {10.1109/CDC.2010.5717742},
year = {2010},
date = {2010-12-15},
booktitle = {Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA},
pages = {690--695},
abstract = {A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. |
Liberzon, Daniel; Trenn, Stephan The Bang-Bang Funnel Controller (long version) Miscellaneous Extended Conference Manuscript, 2010, (long version of corresponding CDC paper). @misc{LibeTren10m,
title = {The Bang-Bang Funnel Controller (long version)},
author = {Daniel Liberzon and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806longVersion.pdf, Long version of corresponding CDC-paper},
year = {2010},
date = {2010-08-06},
abstract = {A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough.},
howpublished = {Extended Conference Manuscript},
note = {long version of corresponding CDC paper},
keywords = {funnel-control, input-constraints, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. |
2006
|
Mandaloju, Nagendra P.; Trenn, Stephan Analogue Implementation of the funnel controller Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 823–824, WILEY-VCH Verlag, 2006, ISSN: 1617-7061. @inproceedings{MandTren06,
title = {Analogue Implementation of the funnel controller},
author = {Nagendra P. Mandaloju and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MT060428.pdf, Preprint},
doi = {10.1002/pamm.200610391},
issn = {1617-7061},
year = {2006},
date = {2006-05-01},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
volume = {6},
number = {1},
pages = {823--824},
publisher = {WILEY-VCH Verlag},
abstract = {In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications.},
keywords = {application, funnel-control, nonlinear},
pubstate = {published},
tppubtype = {inproceedings}
}
In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications. |
2005
|
Ilchmann, Achim; Ryan, Eugene P.; Trenn, Stephan Tracking control: performance funnels and prescribed transient behaviour Journal Article In: Syst. Control Lett., vol. 54, no. 7, pp. 655–670, 2005. @article{IlchRyan05,
title = {Tracking control: performance funnels and prescribed transient behaviour},
author = {Achim Ilchmann and Eugene P. Ryan and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IRT041112.pdf, Preprint},
doi = {10.1016/j.sysconle.2004.11.005},
year = {2005},
date = {2005-01-01},
journal = {Syst. Control Lett.},
volume = {54},
number = {7},
pages = {655--670},
publisher = {Elsevier},
abstract = {Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function
exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.},
keywords = {funnel-control},
pubstate = {published},
tppubtype = {article}
}
Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function
exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary. |
2004
|
Ilchmann, Achim; Ryan, Eugene P.; Trenn, Stephan Adaptive tracking within prescribed funnels Proceedings Article In: Proc. 2004 IEEE Int. Conf. Control Appl., pp. 1032–1036, 2004. @inproceedings{IlchRyan04b,
title = {Adaptive tracking within prescribed funnels},
author = {Achim Ilchmann and Eugene P. Ryan and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IRT040512.pdf, Preprint},
doi = {10.1109/CCA.2004.1387507},
year = {2004},
date = {2004-09-01},
booktitle = {Proc. 2004 IEEE Int. Conf. Control Appl.},
volume = {2},
pages = {1032--1036},
abstract = {Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given lambda > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary.},
keywords = {funnel-control, nonlinear, stability},
pubstate = {published},
tppubtype = {inproceedings}
}
Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given lambda > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary. |
Ilchmann, Achim; Trenn, Stephan Input constrained funnel control with applications to chemical reactor models Journal Article In: Syst. Control Lett., vol. 53, no. 5, pp. 361–375, 2004. @article{IlchTren04,
title = {Input constrained funnel control with applications to chemical reactor models},
author = {Achim Ilchmann and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IT040715.pdf, Preprint},
doi = {10.1016/j.sysconle.2004.05.014},
year = {2004},
date = {2004-01-01},
journal = {Syst. Control Lett.},
volume = {53},
number = {5},
pages = {361--375},
publisher = {Elsevier},
abstract = {Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*.},
keywords = {application, funnel-control, input-constraints},
pubstate = {published},
tppubtype = {article}
}
Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*. |
